ece604 lecture 17

# ece604 lecture 17 - Lecture 17 ECE 604 Linear Systems Doug...

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Nov. 12, 2009 Linear Systems © Douglas Looze 1 Lecture 17 ECE 604 Linear Systems Doug Looze

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Nov. 12, 2009 Linear Systems © Douglas Looze 2 Announcements Problem set 6 available Due Thursday, November 19 MT Exam Average 64
Nov. 12, 2009 Linear Systems © Douglas Looze 3 Last Time Defined concept of realization Connect IO and state variable models Minimal realization Briefly considered time-varying realizations Two SISO time-invariant realizations Controllable Observable No pole-zero cancellations if and only if minimal

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Nov. 12, 2009 Linear Systems © Douglas Looze 4 SISO Realization Given transfer function g ( s ) Find state variable model ( A , b , c , d ) so ( 29 ( 29 1 g s s - = - + c I A b d Assume Ratio of polynomials Strictly proper ( 29 ( 29 ( 29 n s g s d s = ( 29 ( 29 ( 29 ( 29 deg deg n s d s < 0 = d
Nov. 12, 2009 Linear Systems © Douglas Looze 5 If only proper ( 29 ( 29 ( 29 ( 29 deg deg G G n s d s ( 29 ( 29 g s g s = + d Coefficients of numerator and denominator ( 29 , , , is the realization A b c d

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Nov. 12, 2009 Linear Systems © Douglas Looze 6 Controllable Canonical Realization ( 29 ( 29 { ( 29 0 1 2 1 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 1 n d t t u t dt a a a a -       = +       - - - -     b A x x L L M M M M M L L 1 4 4 4 4 4 2 4 4 4 4 4 3 ( 29 [ ] ( 29 0 1 2 1 n y t t γ - = c x L 1 4 4 442 4 4 4 43
Nov. 12, 2009 Linear Systems © Douglas Looze 7 Observable Canonical Realization ( 29 ( 29 ( 29 0 0 1 1 2 2 1 1 0 0 0 1 0 0 0 1 0 0 0 1 n n n a a d a t t u t dt a γ - - - - - - = + - A b x x L L L M M M M M L 1 4 4 442 4 4 4 43 1 2 3 ( 29 [ ] ( 29 0 0 0 1 y t t = c x L 1 4 442 4 443

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Nov. 12, 2009 Linear Systems © Douglas Looze 8 Notes Any basis change will not change (C)/(O) properties Reordering states (Matlab) Minimal if no common factors between numerator and denominator
Nov. 12, 2009 Linear Systems © Douglas Looze

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ece604 lecture 17 - Lecture 17 ECE 604 Linear Systems Doug...

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